arXiv research

Theory proves existence of hypersurfaces with prescribed curvature.

The paper constructs surfaces with prescribed mean curvature in a specific space.

Proves existence of graph on torus with prescribed curvature.

Study on $t$-graphs with prescribed mean curvature in Heisenberg groups.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.

Proves optimal regularity for sphere minimizers in 3-sphere.

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…

Proves existence and uniqueness of Killing graphs with prescribed curvature.

Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type classification hold…

Compact capillary hypersurfaces with specific curvatures are proven.

Study on curvature equation in Heisenberg group with convex boundary.

Survey on rigidity results for graphs with prescribed mean curvature.

Mean curvature flow converges to a translating soliton with prescribed contact angle.

We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.

These are lecture notes for the mini-course \textit{PDE and hypersurfaces with prescribed mean curvature} held in Federal University of São Carlos at the Workshop on Submanifold Theory and Geometric Analysis, August 05 -- 09, 2019. The aim of these notes is to introduce to the geometers useful tools from the \textit{Th…

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

Let $M$ be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function $H$ (with mild conditions) on $M$, we construct a closed incompressible surface with mean curvature $H$ . A direct application is the existe…

Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.

The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

Proves existence of sphere foliations with prescribed mean curvature on Riemannian manifolds.

We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

Study of prescribing scalar curvature and mean curvature on compact manifolds with boundary.

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is t…

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …

Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.

We prove a semi-global result on the existence of conformal embeddings of the two-sphere into the round three-sphere S^3(1) with prescribed mean curvature.

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.

Paper finds unique solutions for curved surfaces with specific gradient.

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

Study translating solitons in Minkowski space with prescribed Gauss image.

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is $f$, when $f$ possesses certain reflection or rotation symmetry.

Paper proves minimizing movements match smooth droplet flow in 3D.

Study minimizes Willmore energy with constraints on surface properties.

New existence results for curvature problem on balls with specific conditions.

Constructs surfaces with specific topologies and curvatures.